Graphing Absolute Value on the TI-83

which will have the value of -1 for all X where
|6X+8|+26 is false, and
the value 2 where that expression is true.

Figure 1

We start this sequence on the Y= screen by pressing
the key. We start with the left parenthesis,
.
In order to insert the abs( we need to move first to the
TEST screen, via the key sequence.

Figure 2

Fortunately for us, the abs( item is the very first in the
CATALOG, as shown in Figure 2. The arrow pointer to the left of the abs( item
could be moved with the up and down cursor keys, but we do not
need to move it at all in this case. We press the
key to select this item and put it into the expression we were creting.

Figure 3

Figure 3 shows the expression with the abs( pasted in from the CATALOG screen.

Figure 4

In Figure 4 we have added more of the expression. We now need to enter the
symbol for "greater than or equal to". To do this we need to move to the
TEST menu via the sequence.

Figure 5

Here is the test menu. The item we want is item number 4.
We need only press the to select the item, close the
screen and insert the item into the expression.

Figure 6

Figure 6 confirms the insertion of the symbol.

Figure 7

We finish the expression in Figure 7.
Each time this expression is evaluated it will produce a 0 if it is false, and a
1 if it is true. We wanted to change those values so that the graph does not
fall on the X-axis.
The opening and closing parentheses allow us to multiply the
expression by 3 and then we can subtract 1.
This will change false values to -1 and true values to 2.

Figure 8

Figure 8 shows the completed expression. We are ready to
graph the expression. To do this we press the key.

Figure 9

Here is the graph of the solution.
We can see that the expression is true everywhere except from about -2 to about -2/3.
We can focus in on this region by changing the WINDOW settings.
To do this we press the key.

Figure 10

Figure 10 shows the WINDOW settings after we have changed them to
concentrate our graph on the region from -2.5 to 0.5.
At the same time we have changed the Y-values to range from -4 to 6.

Figure 11

Pessing the key returns us to the
GRAPH screen shown in Figure 11.

Figure 12

For Figure 12 we have pressed the
key to move into the TRACE mode of the calculator, and we
have used the key to move to the
right end of the lower segement of the graph. Note that the X-value
is -.6808511. If we had moved one more pixel to the right we would
jump up to the upper portion of the graph, and the X-value would be
greater than -2/3. If we wanted to, we could return to the WINDOW settings and change them
so that we examine just the region from -.68 to -.66, giving us an even
better reading of the X-values at the ends of the upper and lower
segments. For this example we will not do this. The graph here is meant to
confirm, to check, the algebraic solution in the text.

Figure 13

We can use the key to
move the TRACE pointer to the left. In Figure 13 we have moved it until it has jumped to the
left upper portion of the graph. Again, we can see the X-value of the
particular pixel.

The final three figures on this page are meant to demonstrate the
relation between what we say we are doing, and what we are really doing.
The original problem was to solve

|6x+8|+26

Looking at this problem, here or in the text, we might notice that the variable Y
does not appear in the problem. The original problem is an inequality in one variable, X.
In the Figures above, we have worked to create a two-level graph of this problem, showing where the
inequality is true and where it is false. We really created a two variable function, namely,

Y1=(abs(6X+8)+26)*3-1

Now we will look at the related problem

|6x+8|+2-60

where we have simply moved the 6 to the left side of the
sign, giving us an expression on the left that is
greater than or equal to 0. If we look at

Y2=abs(6X+8)+2-6

then we are really asking, where is Y2 greater than or equal to 0.

Figure 14

In Figure 14 we have returned to the Y= screen and we have entered the
new, unsimplified, equation.

Figure 15

We return to the GRAPH screen and we can see not only the original two level
function, but also the graph of our new function

Y2=abs(6X+8)+2-6

which comes out as a "V". Notice that the "V" is at or above the X-axis
for all X-values where the two level function is at the upper level, i.e., is true.
And, the "V" is below the X-axis for those values of X where the two level function
is at -1, i.e., is false.
In order to better see this, Figure 15 has been augmented and reproduced as Figure 16.

Figure 16

The change from Figure 15 to Figure 16 is that we have drawn two red and
two blue vertical lines.
The red lines are above and below the endpoints of the lower level segment of the two level
function. That is, they mark the left and right ends of the region where the original
expression was false. The blue lines are above and below the endpoints of
the upper level segments of the two level function.
That is, they mark the right and left ends of the region where the original
expression was true.

These red and blue lines were added to make it easier to
see where the "V" graph of the second function is above the X-axis and where it is
below the X-axis.